General Linear Group Action on Tensors: A Candidate for Post-quantum Cryptography

Abstract

Starting from the one-way group action framework of Brassard and Yung (Crypto '90), we revisit building cryptography based on group actions. Several previous candidates for oneway group actions no longer stand, due to progress both on classical algorithms (e.g., graph isomorphism) and quantum algorithms (e.g., discrete logarithm).We propose the general linear group action on tensors as a new candidate to build cryptography based on group actions. Recent works (Futorny-Grochow-Sergeichuk Lin. Alg. Appl., 2019) suggest that the underlying algorithmic problem, the tensor isomorphism problem, is the hardest one among several isomorphism testing problems arising from areas including coding theory, computational group theory, and multivariate cryptography. We present evidence to justify the viability of this proposal from comprehensive study of the state-of-art heuristic algorithms, theoretical algorithms and hardness results, as well as quantum algorithms.We then introduce a new notion called pseudorandom group actions to further develop groupaction based cryptography. Briefly speaking, given a group G acting on a set S, we assume that it is hard to distinguish two distributions of (s, t) either uniformly chosen from S × S, or where s is randomly chosen from S and t is the result of applying a random group action of g ∈ G on s. This subsumes the classical decisional Diffie-Hellman assumption when specialized to a particular group action. We carefully analyze various attack strategies that support the general linear group action on tensors as a candidate for this assumption.Finally, we establish the quantum security of several cryptographic primitives based on the one-way group action assumption and the pseudorandom group action assumption.

Cryptographic Group Actions and Applications

“…Cryptographic group actions have received substantially less attention compared to traditional group-theoretic assumptions. Nonetheless, there have been a small number of works studying various candidate cryptographic group actions [GS10,JQSY19] and their hardness properties [BY91,GPSV18]. In terms of public-key primitives, these works have demonstrated that cryptographic group actions endowed with some hardness properties imply PKE and noninteractive key exchange (NIKE).…”

Pembuktian Sifat non-Hausdorff dari Grup Lie GL(n,C) Bertindak pada M(n,C)

“…Grup General Linear pada tensor yang merupakan bentuk lebih umum dari , ℂ juga masih menjadi objek yang masih terus diteliti, hal ini ditunjukan dengan adanya paper [4] yang juga membahas aplikasi tindakan grup general linear pada kriptografi. Tindakan grup yang diperoleh digunakan untuk membangun suatu skema untuk enkripsi public-key.…”

From the Hardness of Detecting Superpositions to Cryptography: Quantum Public Key Encryption and Commitments

“…Cryptographic group actions are recently gaining a renewed attention from the perspective of postquantum cryptography. Ji et al [JQSY19] proposed new instantiations based on general linear group We remark that the meaning of "swap" here is different from that of the SWAP gate as explained below. The distinguishing advantage is (necessarily) halved.…”

“…An important difference between the instantiations in [JQSY19] and [ADMP20] is that the former considers non-abelian groups whereas the latter considers abelian groups. Abelian group actions are particularly useful because they give rise to a non-interactive key exchange protocol similar to Diffie-Hellman key exchange [DH76].…”

“…On the other hand, the above construction does not work if G is a non-abelian group. Indeed, cryptographic applications given in [JQSY19] are limited to Minicrypt primitives [Imp95], i.e., those that do not imply PKE in a black-box manner. Thus, [JQSY19] raised the following open question:…”